Confluent and Double-Confluent Heun Equations: Convergence of Solutions in Series of Coulomb Wavefunctions

摘要

The Leaver solutions in series of Coulomb wave functions for the confluentHeun equation (CHE) are given by two-sided infinite series, that is, by serieswhere the summation index $n$ runs from minus to plus infinity [E. W. Leaver,J. Math. Phys. 27, 1238 (1986)]. First we show that, in contrast to theD'Alembert test, under certain conditions the Raabe test assures that thedomains of convergence of these solutions include an additional singular point.Further, by using a limit proposed by Leaver, we obtain solutions for thedouble-confluent Heun equation (DCHE). In addition, we get solutions for theso-called Whittaker-Ince limit of the CHE and DCHE. For these four equations,new solutions are generated by transformations of variables. In the secondplace, for each of the above equations we obtain one-sided series solutions($n\geq 0$) by truncating on the left the two-sided series. Finally we discussthe time dependence of the Klein-Gordon equation in two cosmological models andthe spatial dependence of the Schr\"{o}dinger equation to a family ofquasi-exactly solvable potentials. For a subfamily of these potentials weobtain infinite-series solutions which converge and are bounded for all valuesof the independent variable, in opposition to a common belief}.